WEIGHTS AND MEAN ERRORS. 



ccliii 



another. This may readily and conveniently be done by introducing an empirical constant 

 of the form A = fXTi i n the pl ac e of b. Then, 



n A(n 1) A-\-n(l A) 



The annexed table is based upon the hypothesis of A =. |- ; or, what is the same thing, 

 b =. 5, a value which experiment appears to indicate as more probable than any other integer 

 for the average of the observations under discussion. 



Weights as Functions of the Number of Observations. 



Where two results are to be combined, the one derived from n, the other from n observations, 

 and having respectively the weights P (n) and p( n ) } we shall have 



P&amp;lt;*&amp;gt;* 



(n ) 



or 



And since 



= n. 



n,n ) 



p ( n ) __ 



p(n) I p(n&amp;gt;) 

 &amp;lt; 1+* 



P(n,n ) 



1 + 6 



2 n n + (n -f n ) b 



nn 



Or 



p(n,n&amp;gt;) 



1+6 



From this formula, and the assumption as before b = 5, the table is constructed which gives 

 the weights as functions of two numbers of observations, the decimal point being avoided by 

 assuming 100 as the unit of weight. The quantities under discussion being the means of the 

 two sets of observations, are entitled, after halving, to a double weight, so that the tabulated 

 function is 



O 1Dfn.nl} 1 HA 



The last table contains, of course, the values given by the present one for the special case 

 when n= n ; that is to say, when the number of observations is equal in the two sets and their 

 mean is taken. 



